arXiv research

Unified diffusive bounds for non-linear parabolic equations.

Study shows long-term solutions for complex equations on curved spaces.

Extends parabolic study to flat hyperkähler manifolds.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

Solves pair trading problem using consumption-investment theory.

I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservatio…

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quan…

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of our result is the fact that, besides of obtaining a…

The moving coframe method is applied to solve the local equivalence problem for the class of linear parabolic equations in two independent variables under an action of the pseudo-group of contact transformations. The structure equations and the complete sets of differential invariants for symmetry groups are found. The…

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models …

Study asymptotically almost periodic solutions on real hyperbolic manifolds.

The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.

We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, th…

Smooth solutions up to evolving free boundaries for degenerate equations.

In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary…

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on Kähler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Ampère equations when the background Kähler metrics on $\mathbb{C}^n$ have cone singul…

Study deep neural nets for solving complex insurance equations.

This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the Hölder-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has time-derivative bounded from above. On a Kähler manifold, this Hölder estimate works when the …

The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.

Fiedler and Mallet-Paret prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapuno…

Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space $H_1$. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Mar…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

We study a method of reducing space dimension in multi-dimensional Black-Scholes partial differential equations as well as in multi-dimensional parabolic equations. We prove that a multiplicative transformation of space variables in the Black-Scholes partial differential equation reserves the form of Black-Scholes part…

This is the continuation of our paper \cite{GS}, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, …

Study on slow convergence in geometric variational problems.

Within a financial model with linear price impact, we study the problem of hedging a covered European option under gamma constraint. Using stochastic target and partial differential equation smoothing techniques, we prove that the super-replication price is the viscosity solution of a fully non-linear parabolic equatio…

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

Sharp estimates for parabolic equations on manifolds using symmetrization.

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at infinity. We connect the behavior of non-negative solutions to the interplay betwe…

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Second part of a study on heat equations on special manifolds, focusing on parametrix construction.

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We …

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.

New method for analyzing elliptic and parabolic equations.

Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Di…

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This…

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.

Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data.

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.